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# Pendulum rocket fallacy

The pendulum rocket fallacy is a common fundamental misunderstanding of the mechanics of rocket flight and how rockets remain on a stable trajectory.

Many liquid fuel rockets constructed by the early pioneers of rocketry in the 1920s and 1930s differed significantly from modern rockets in that the rocket engine was placed at the top and the fuel tank at the bottom of the rocket. The first liquid-fueled rocket, constructed by Robert Goddard, was typical of this.

It was believed that, in flight, the rocket would "hang" from the engine like a pendulum hanging from a pivot, and the weight of the fuel tank would be all that is needed to keep the rocket flying straight up. However, this belief is incorrect – such a rocket will turn and crash into the ground soon after launch. This is what happened to Goddard's rocket. In fact it can be shown using basic Newtonian mechanics that the rocket is exactly as unstable as when the rocket engine is mounted under the fuel tank, as is the case in most modern rockets.

## Practical explanation

No rocket can be perfectly constructed. Inevitably, the engine's direction of thrust will be imperfectly aligned with the rocket's axis so the rocket will have a slight inbuilt tendency to turn. When this happens, the engine starts rotating with the rest of the rocket, regardless of its shape, and the direction of thrust rotates as well.

Except air resistance, there is no rotational force or torque available to turn the rocket back onto its correct path, as can be shown from the classical Newtonian physics reasoning in the next paragraph. As a consequence, the initial deviation from a vertical path will increase over time, and a rocket constructed in this way will always turn around and strike the ground sooner or later.

## Physical reasoning

The pendulum belief is a fallacy because it stems from the implicit (and false) assumption that simply because the weights and "hanging" devices are arranged in roughly the same way in both the rocket and the pendulum, they will behave in the same fashion. However, the forces exerted are different. While gravity does act similarly in both physical systems, the supporting force exerted onto the pendulum by its hanging point is constrained to remaining aligned with said fixed point; this is unlike the force exerted onto the rocket by its engine, whose direction instead depends on the rocket's overall orientation or attitude.

Consider the physical system constituted by a rocket like Goddard's, comprising the engine, tank, and rigid frame. Assuming that air resistance can be neglected, there are only two forces that are exerted on the system as a whole: gravity, and the reaction force that is caused by the ignited gases being expelled from the rocket's nozzle at high speed. Let's examine the moment of each of these forces with respect to the center of mass of the system. Gravity : The center of gravity is identical to the center of mass and therefore gravity does not exert any torque. This is a general property of all systems in a uniform gravitational field. Reaction force from the engine : Due to the rigid construction of the rocket frame, the force is exerted on a line that is fixed with respect to the rocket. The unavoidable imperfection mentioned above means that this line does not contain the center of mass exactly. The amplitude of the reaction force depends on the thrust of the engine, which is always positive. Therefore, the torque is exerted with respect to an axis whose direction is fixed with respect to the rocket frame, and is of constant sign.

Given that torques are pseudo-vectors and hence add linearly, it follows that the rotation speed of the rocket around the aforementioned axis can only increase.

## Solutions

To fly correctly, rockets must have another means of stability. The fins of model rockets and the sticks of firework rockets act aerodynamically to keep the axis of the rocket pointing in the direction of flight. Larger rockets can do without fins by using a guidance and control system that actively steers the rocket and keeps it flying in the intended direction.

Even a Goddard-type rocket, with the engine at the front, will fly correctly if fitted with fins or another means of control. An example of this are the launch escape systems fitted to some crewed spacecraft. These are aerodynamically stable; indeed, in the case of the Apollo spacecraft, engineers had to fit several hundred kilograms of depleted uranium ballast to the nose of the escape rocket in order to move the center of gravity far enough forward.

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